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Submitted on 5 May 2015

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Jules Morand

To cite this version:

Jules Morand. Dynamics of long range interacting systems beyond the Vlasov limit. High Energy Physics - Theory [hep-th]. Université Pierre et Marie Curie - Paris VI, 2014. English. �NNT : 2014PA066624�. �tel-01148696�

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Thèse de Doctorat

pour obtenir le grade de docteur ès sciences

Spécialité : Physique statistique - Systèmes complexes préparée au

Laboratoire de Physique Nucléaire et Hautes Énergies dans le cadre de l’École Doctorale ED 389

présentée et soutenue publiquement par

Jules Morand

le 2 décembre 2014 Titre:

Dynamics of long range interacting systems

beyond the Vlasov limit.

Dynamique des systèmes à longue portée

au delà de la limite de Vlasov.

Directeur de thèse: Michael Joyce

Jury

M. Angel ALASTUEY, Examinateur M. Julien BARRÉ, Rapporteur

M. Michael JOYCE, Directeur de thèse Mme. Régine PERZYNSKI, Examinatrice

M. Stefano RUFFO, Rapporteur M. Emmanuel TRIZAC, Examinateur M. Pascal VIOT, Invité

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Contents . . . iii

Introduction 1 1 Introduction to long-range interacting systems 3 1 Thermodynamics and dynamics . . . 3

1.1 Definition . . . 3

1.2 The distinctive thermodynamics of long-range systems . . . . 4

1.3 Dynamics of long-range systems . . . 9

1.4 Long-range systems with stochastic perturbations . . . 16

2 Self-gravitating systems . . . 17

2.1 3d self-gravity . . . 17

2.2 1d self-gravity . . . 20

2 Finite N corrections to Vlasov dynamics 25 1 A Vlasov-like equation for the coarse-grained phase space density . . 27

2 Statistical evaluation of the finite N fluctuating terms . . . 30

2.1 Mean and variance of ⇠v . . . 31

2.2 Mean and variance of ⇠F . . . 31

3 Parametric dependence of the fluctuations . . . 35

3.1 Mean field Vlasov limit . . . 35

3.2 Velocity fluctuations . . . 36

3.3 Force fluctuations . . . 36

4 Force fluctuations about the Vlasov limit: dependence on " . . . 38

4.1 Case < d 2 . . . 38 4.2 Case d 2 < < d + 1 2 . . . 39 4.3 Case d +1 2 < . . . 40

5 Exact one dimensional calculation and numerical results . . . 40

6 Discussion and conclusions . . . 45

7 Appendix 1: Alternative derivation of h⇠2 Fi . . . 48

3 Long-range systems with weak dissipation 51 1 Introduction . . . 51

2 Mean field limit for long-range systems with dissipation . . . 52

2.1 Dissipation through viscous damping . . . 52

2.2 Dissipation by inelastic collisions. . . 54

3 Scaling quasi-stationary states . . . 57

4 Numerical study: 1d self-gravitating system . . . 62

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4.2 VDM – 1d self-gravity with viscous damping force . . . 65

4.3 ICM – 1d self-gravity with inelastic collisions . . . 67

4.4 Initial conditions for numerical simulation . . . 68

5 Simulation results for 1d self-gravity . . . 73

5.1 Macroscopic observables . . . 73

5.2 SGS (without dissipation) . . . 74

5.3 VDM: SGS with viscous damping . . . 80

5.4 ICM: SGS with dissipation through inelastic collisions . . . 83

6 Conclusion . . . 92

4 Long-range systems with internal local perturbations 95 1 Introduction . . . 95

2 The stochastic collision model (SCM) . . . 97

2.1 Microscopic description of the stochastic collisions . . . 97

2.2 Kinetic equation . . . 98

2.3 Kramers Moyal expansion of the collision operator . . . 99

2.4 Moments of the collision operator . . . 100

2.5 Temporal evolution of moments . . . 101

2.6 Instability of thermal equilibrium . . . 102

2.7 Evolution of energy . . . 103

2.8 Numerical results for the granular gas . . . 106

2.9 Numerical results for 1d self-gravitating gas . . . 110

2.10 Conclusion on SCM . . . 117

3 The BSR model . . . 118

3.1 BRS collisions . . . 118

3.2 Numerical results for BRS model without gravity . . . 120

3.3 Numerical results: BRS with gravity . . . 124

3.4 Conclusion on BRS collisions. . . 132

4 Conclusion . . . 132

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Interactions are canonically characterized as short-range or long-range on the ba-sis of the fundamental distinction which arises in equilibrium statistical mechanics between interactions for which the energy is additive and those for which it is non-additive. For a system of particles interacting via two body interactions with a pair potential V (r), the system is then long-range if and only if V (r) decays at large dis-tances slower than one over the separation r to the power of the spatial dimension d. In the last decade there has been considerable study of this class of interactions (for reviews, see e.g. [1, 2]). One of the very interesting results about systems in this class which has emerged — essentially through numerical study of different models — is that, like for the much studied case of gravity in astrophysics, their dynamics leads, from generic initial conditions, to so-called quasi-stationary states (QSS): macroscopic non-equilibrium states which evolve only on time scales which diverge with particle number. Their physical realizations arise in numerous and very diverse systems, ranging from galaxies and “dark matter halo” in astrophysics and cosmology (see e.g. [3]), to the red spot on Jupiter [4], to laboratory systems such as cold atoms [5], and even to biological systems [6]. Theoretically these states are commonly interpreted in terms of a description of the dynamics of the system by the Vlasov equation of which they represent stationary solutions. The non linear Vlasov equation describes the evolution of a smooth one particle distribution, in the limit of a large system N ! +1. In this limit, the Vlasov equation is exact. For finite systems, the Vlasov equation only describe its evolution up to a finite time diverging with the system size. These results are valid for conservative system in the micro-canonical framework.

In nature, however, real systems like those mention above, are in general finite, not isolated, and the entities (or particles) that compose the system are macroscopic, have internal degrees of freedom with short range interactions which might not be conservative. Thus the question naturally arises, is whether the presence of such QSS is valid and more generally, how the dynamics of long-range system is modi-fied, beyond the idealised limit of Vlasov. The questions at the center of the thesis are the following: Under what conditions on a pair interaction does one expect the QSS to exist for isolated (Hamiltonian) systems? Do we expect QSS to exist also when such Hamiltonian system is subjected to additional dissipative forces? Do we expect QSS to exist in the presence of small stochastic perturbations to the system? In the first chapter, we briefly introduce some essential results about the thermo-dynamics and thermo-dynamics of long-range system and in particular, self-gravitating sys-tems. We present notably a derivation of the Vlasov equation through the BBGKY hierachy and discuss its limitations.

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In a second chapter, we explore the conditions on a pair interaction for the va-lidity of the Vlasov equation to describe the dynamics of an interacting N particle system in the large N limit. Using a coarse-graining in phase space of the exact Klimontovich equation for the N particle system, we evaluate, neglecting correla-tions of density fluctuacorrela-tions, the scalings with N of the terms describing the correc-tions to the Vlasov equation for the coarse-grained one particle phase space density. Considering a generic interaction with radial pair force F (r), with F (r) ⇠ 1/r at large scales, and regulated to a bounded behaviour below a “softening” scale ", we find that there is an essential qualitative difference between the cases < d and > d, i.e., depending on the the integrability at large distances of the pair force. In the former case, the corrections to the Vlasov dynamics for a given coarse-grained scale are essentially insensitive to the softening parameter ", while for > d the amplitude of these terms is directly regulated by ", and thus by the small scale properties of the interaction. This corresponds to a simple physical criterion for a basic distinction between long-range (  d) and short range ( > d) interactions, different to the canonical one (  d + 1 or > d + 1 ) based on thermodynamic analysis. This alternative classification, based on purely dynamical considerations, is relevant notably to understanding the conditions for the existence of so-called quasi-stationary states in long-range interacting systems. This chapter follows very closely the content of an article submitted to Physical Review E [7].

In a third chapter, which is an extended report on the study of which the high-light results were published in Physical Review Letters [8], we consider long-range interacting systems subjected to additional dissipative forces. Using an appropriate mean-field kinetic description, we show that models with dissipation due to a vis-cous damping or due to inelastic collisions admit “scaling quasi-stationary states”, i.e., states which are quasi-stationary in rescaled variables. A numerical study of one dimensional self-gravitating systems confirms both the relevance of these solutions, and gives indications of their regime of validity in line with theoretical predictions. We underline that the velocity distributions never show any tendency to evolve towards a Maxwell-Boltzmann form.

In the last chapter, which is a report on work in progress (and not yet published), we study two different toy models to explore how different kinds of stochastic per-turbation affect the dynamics of long-range interaction system, and, in particular, the QSSs which are characteristic of them. In the models used, both extensions of the model with inelastic collision of the previous chapter, the perturbation is asso-ciated with the collision events, which can lead to gain or a loss of kinetic energy. For a first model, the collision are built such that on average over the realizations, the total energy in conserved, and in a second model, they are such that the system relax to a state where the loss and the gain of energy balance. For both models the perturbation drives the system to evolve, through a family of QSS, into a non-equilibrium stationary state. This final stationary distribution, which is itself also a stable QSS, does not depend on the initial conditions but does depend strongly on the microscopic detail of the perturbation.

Thus our conclusion is that QSS appear to be a very robust feature of long-range systems, and their existence or occurrence is not conditioned in general on the idealization that the system is isolated.

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Introduction to long-range

interacting systems

1 Thermodynamics and dynamics

1.1 Definition

A long-range interacting system is usually defined as collection of entities (particles, spins, vortices,..) interacting with a pair potential which is not integrable at r ! +1. As we explain below, these systems have the property of non additivity and this property has strong consequences on the equilibrium statistical mechanics of the system.

To be a little more precise, and explain the origin of this definition let us estimate, in d dimensions, the potential energy ep of a particle placed at the center of sphere

of radius R enclosing a volume V = ⌦dRd where other particles are homogeneously

distributed (where ⌦dis the appropriate constant for the dimension d). We consider

that particles interact with a power low central potential defined as (r) = g

|r|↵, (1.1)

where g is the coupling constant of the model and ↵ the exponent characterizing the interaction. We use a cut-off ", to regulate the possible short-scale divergence of (for ↵ > 0) i.e. we do not consider the contributions to ep from particles located

within a sphere of small radius " around the particle considered. We will return below to this point to discuss further the role and necessity for this cut-off. In the continuous limit, the individual potential energy is expressed as:

ep = Z R " ddrn0 g r↵ =0 J⌦d Z R " drr1 = n0g⌦d d ↵(R d ↵ "d ↵) , if ↵6= d

where n0 = NV is the particle density. The following distinction then follows.

• For ↵ > d, the potential is integrable: the energy per particle remains finite if we take R ! +1 at fixed n0 (i.e. the usual thermodynamic limit). In this

limit, the contribution to ep due to far away particles is negligible compared

to that due to the ones close to it. Further, the total energy, E =RV ddrn 0ep,

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is extensive: it scales linearly in the thermodynamic limit with the volume V . For such system, the energy per particle is an intensive quantity and it is well defined in thermodynamic limit.

This class of system, which we will refer to, following the canonical usage, as short-range interacting system, includes most of the familiar systems studied in physics, and in particular those usually considered in the context of the study of phase transitions and critical phenomena. In this scope one usually defines the exponent of the potential with the parameter = ↵ d, and the studies are usually restricted to the case > 0. To avoid ambiguity, we mention that in this latter community the term “long-range” is then often used to qualify a universality class of systems for which the critical exponents characterizing the divergence of physical quantities, affected by the nature of the interaction, are different to the one of the mean field limit and induce long-range correlations [2, 9]. This typically happens in the long-range d/2 < < (d) where (d) is a critical exponent for which the precise d-dependence (in d  4) is still discussed [10, 11].

• For ↵ 6 d, the potential energy per particle (or density of potential energy) ep diverges for R ! +1 in the usual thermodynamic limit. The potential is

non-integrable. This is by definition the case of long-range interacting systems

1. One cannot neglect the contribution of e

p due to “far-away” particles. The

energy per particle scaling as ep / Rd ↵ / V1 ↵/d, is no more intensive and

the total potential energy of the system U / V2 ↵/d is super-extensive: it

increases more rapidly than linearly with the volume.

The long-range interactions concerns a large variety of systems [12] such as plasma [13], self-gravitating systems [3,14], 2d turbulent flow [15], cold atoms [5,16], biological systems [6]. Further, many theoretical toy models such as the Hamitonian mean field (HMF) model(e.g. [1] and reference therein), the Blume-Emery-Griffiths (BEG) model [17], the 1d self-gravitating [18–20] or self-gravitating ring model [21] have been proposed to study the physics of long-range systems.

1.2 The distinctive thermodynamics of long-range systems

Various crucial differences with respect to short-range systems emerge immediately when one applies equilibrium statistical mechanics to long-range systems.

The first and fundamental one is that related to the definition given above. When we study the equilibrium thermodynamics of any system we must have extensivity of the total energy E as N ! +1 and V ! +1 (in order to have an N independent energy density). In order to recover this property, we must define the thermodynamic limit differently. There are various mathematically equivalent ways of doing this. One, the so-called Kac prescription, is to scale the coupling constant g with the size of the system as V (1 ↵/d) with the density n

0 = NV is fixed, and the coupling

constant decreases in increasing the system size.

Alternatively, in the context of 3d self-gravitating system (for which g = Gm2)

one can use the “dilute” limit with g / 1

N2 and n0 / V (1+d). Recently, another

1or “strong” long-range system in opposition to the “weak” long-range interaction to distinguish

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scaling limit have been proposed in [22], the author propose a scaling limit for a self-gravitating system of hard spheres keeping constant the packing fraction.

Such a procedure restores the extensivity of the system; but nevertheless it re-mains non-additive i.e. the interaction energy of any part of the system with the whole is not negligible with respect to the internal energy of the given part. We detail further in the next section this point and discuss its particular consequence on the thermodynamics of long-range interacting systems.

A second issue which present itself immediately in analysing the equilibrium ther-modynamics of long-range systems is one which occurs for any attractive power law potential 1/r↵ with ↵ > 0. An interesting example, of this class of system includes,

3d self-gravitating system whose thermodynamics is discussed further below. Just as for short-range systems whose exponents are also positive, such systems require a regularisation at short distances to avoid collapse. Indeed if only two particles get closer and closer one another, these two can make the potential energy of the entire system diverges to 1. The total energy being conserved, the kinetic energy of the system increases and the number of accessible states diverges in the velocity space. As a consequence, the micro-canonical partition function:

⌦(E)/ Z

ddNxddNx (E H({xN, vN})), (1.2)

which enumerates the number of microscopic states at a fixed energy E, diverges if and only if N 3 [23]. Moreover, the Boltzmann entropy

S(E) = kBln(⌦(E)) (1.3)

increases without bound as well; and without entropy maxima an equilibrium state cannot exist.

Therefore the introduction of a short-scale regulator is needed to bound below the potential energy. For spin systems defined on a lattice like the BEG models, this divergence is naturally regulated. For particle system, one may also consider fermions and made use of the Pauli exclusion principle as it a been done for a self-gravitating system [24], or again hard core particle as in [22]. In general, an ad-hoc cut-off is added to bound the potential. We will discuss in chapter 2, how the large scale dynamics of a long-range system is affected by the choice of the small scale regularisation.

Finally, just as for systems, the number of accessible states (Eq. (1.2)) will also diverge if the system is not confined, unless the pair potential is attractive enough to “confine” the particles (as is the case, as we will see, for example for 1d self-gravitational system). Thus, one has often to impose periodic boundary condition, or confine the system in a box.

Non-additivity

The central difficulty, as we have underlined, characteristic of any long-range system, arises from the non-integrable nature of the long-range potential at large distance. If the Kac prescription restore the extensivity of the long-range interacting system

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Figure 1.1: Schematic picture of a system partitioned into two parts, in red is the boundary between the two subsystems.

and makes possible the existence of a thermodynamic limit, such system, however, remains non-additive. Let us consider first a short-range system, and partition the system in two subsystems 1 and 2 (as in Fig. 1.1) of respective energies E1 and

E2. The total energy of the system is given by E = E1 + E2 + E12 where E12 is

the energy of interaction between the two subsystems. For a short-range system, a particle, situated close to the boundary between the two subsystems, interacts only with particles close to the other side of the boundary, while the interactions with the particle in the bulk of the other subsystem are negligible. Therefore, the energy of interaction E12is proportional to the surface of contact between the two volumes.

Then, in thermodynamic limit E12 becomes negligible with respect to E1 and E2

and in this limit E = E1+ E2: the system is additive.

On the contrary for a long-range system, any particle in one of the two subsystem interacts with every single particles of the other subsystem. The energy of interaction E12 is no longer negligible at the thermodynamic limit and E 6= E1+ E2.

Ensemble inequivalence

An important result of the statistical mechanics built for short-range systems is the equivalence of statistical ensemble which means that mean quantities converge to the same value in the thermodynamic limit. Indeed this result allows one, for example, to study the phase diagram of a system in the ensemble of its choice. Short-range interacting systems are usually study in the canonical ensemble, because calculations are easier than in the micro-canonical ensemble. The latter is essential, however, at a fundamental level to construct the canonical ensemble [1, 12, 25]. We now briefly recall the latter to highlight how the distinction on the basis of additivity is important between short-range and long-range interacting systems.

One considers an isolated system with energy E and divides it into two parts: a “small” part of energy E1 that will be the system we want to study the statistics

and a “large” part of energy E2 E1 playing the role of the thermal bath. The

probability that the system 1 has an energy E1, for an additive system is given by

p(E1) = 1 ⌦(E) Z (E1+ E2 E)⌦2(E2)dE2 (1.4) = ⌦2(E E1) ⌦(E) (1.5)

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where ⌦2is the micro-canonical partition function of the subsystem defined similarly

than ⌦ in Eq. (1.2). Then, using the expression of the micro-canonical entropy (Eq. (1.3)), and Taylor expanding with E >> E1, we have:

p(E1) = 1 ⌦(E)e 1 kBS(E E1) (1.6) = 1 ⌦(E)e 1 kBS(E)+ 1 kBE1 @S @E|E+o(E 2 1) (1.7) ' 1 Ze E1, (1.8)

with Z the partition function and where we have define the inverse temperature of the bath:

= 1 kB

@S

@E E. (1.9)

We thus recover the canonical distribution for the system 1. We clearly see (Eq. (1.4)) that the property of additivity is crucial to build the canonical ensemble. For a short-range system, of course, this construction become strictly valid at the thermodynamic limit. For a long-range system, this construction is not mathemat-ically valid even at the thermodynamic limit. Hence non additive systems, and in particular long-range one, are expect to have a very particular behaviour when they are in contact with a thermal reservoir. The micro-canonical ensemble is clearly well defined and the only issue is how to take the limit N ! +1. There is no formal barrier to calculate, for long-range system, in the canonical ensemble, but its physical meaning must be scrutinized carefully. For many long-range model, the calculation have been done in both ensembles, defining formally the free energy2

and the canonical partition function.

For some long-range systems, it turns out that the equilibrium results in both ensembles are actually equivalent. This is, for example, the case of the 1d self-gravitating system [20, 26, 27], or of the HMF model [1]. For these two models, the thermal equilibrium can be work out in both ensembles and the equilibrium statistical mechanics is well defined. Moreover the dynamical studies have shown that the two long-range models effectively relax to the predicted equilibria which has, in particular, a Gaussian velocity distribution [1, 28].

However for other long-range systems, like in 3d self-gravity [27, 29] or for BEG model [17], the calculation shows that the ensembles are not equivalent: the equilib-rium solution found in maximising the entropy in the micro-canonical ensemble, or in minimizing the free energy in the canonical ensemble, are different in a substantial region of the space of the parameters.

Non concave entropy and negative specific heat

It has been understood and shown rigorously [17,30] that the fundamental explana-tion of the non-equivalence of ensemble is the presence of a non-concave region in the entropy-energy curve in the micro-canonical ensemble.

2We note that the Kac prescription (or other equivalent) to make the energy extensive is

im-portant to define the free energy F = E T S. Indeed as the entropy S / N, one needs E / N so that both term defining F have the same scaling with N.

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Figure 1.2: Example of a concave entropy-energy curve (left panel) and its Legendre-Fenchel transform (right panel): the free-energy temperature curve.

To be more precise, the thermodynamic potential in the micro-canonical ensem-ble (S(E)) and the one in the canonical ensemensem-ble (the free energy F ( )) are related by a Legendre-Fenchel transform [31]. We denote this operation with the subscript ⇤. The entropy S and the free-energy F are thus related by:

S⇤( )= F ( ) = inf. E{ E S(E)} (1.10)

Similarly one defines the Legendre transform of the free-energy which is the canonical entropy:

Scan= S. ⇤⇤(E)= ( F ). ⇤(E) = inf { E F (E)} (1.11) In points where the curve are differentiable, this transformation implies the re-lation

= @S

@E (1.12)

which is the micro-canonical definition of the temperature.

In Fig. 1.2,3one can see that when the entropy is a concave function of the energy

E, the Legendre-Fenchel transform is self-inverse. This means that if one considers the free-energy F ( ) which is the thermodynamic potential in the canonical ensemble and that one wants to go back in the micro-canonical ensemble, using a Legendre transform, then the canonical entropy Scan= Fequals the micro-canonical one S.

In this case then, the ensembles are equivalent.

In Fig. 1.3, we see that this is not the case for a non-concave entropy. This gives rise to a non-differentiable point is the transformed function S⇤ = F (middle

panel). Moreover when one wants to transform the free energy into an entropy, the function obtained, is the concave envelope of the original entropy and in this case the Legendre-Fenchel transform is not self-inverse: S⇤⇤6= S (right panel).

Hence, for systems with a convex intruder in the entropy-energy curve, the en-sembles are not equivalent. For such systems, the micro-canonical ensemble may show richer behaviour than the canonical one. For example, considering the specific heat:

Cv =

@E

@T, (1.13)

3Figures taken form the slides of Hugo Touchette: http://www.maths.qmul.ac.uk/~ht/

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Figure 1.3: Sketch of a non concave entropy S verus the energy E (left panel), its Legendre transform S⇤( ) = F ( )(middle panel), and the Legendre-Fenchel

trans-form of the free-energy (full line, right panel) F⇤(E) = S⇤⇤(E)6= S(E).

related to the entropy by the relation @2S @E2 = 1 T2 1 Cv . (1.14)

In the region where the curve S(E) is convex, the second derivative is positive, and then Cv < 0. A counter intuitive consequence is that when ones increase the

energy of the system, it may cool down. Moreover non-concave entropy is related to a rich variety of phenomena such as the appearance of first-order phase transitions as well as metastable states in the canonical ensemble [32] and also the possibility of ergodicity breaking [33]. In [32] the authors propose a complete classification of micro-canonical phase transitions in long-range interacting, their link to canonical ones, and of the possible situations of ensemble non-equivalence.

The results from equilibrium statistical mechanics aim to predict the final macro-scopic state of the system. However, to determine if the system does effectively relax to such states, the dynamics of the system has to be studied.

1.3 Dynamics of long-range systems

Qualitative description

In a long-range interacting, the contribution to the force acting on a given particle cannot be approximated by that due to particles in its neighbourhood. The re-sultant strong coupling of numerous degrees of freedom allows important collective effects in the many body dynamics. The detailed motion of every single particle is, in practice, inaccessible except by numerical simulation, and what we attempt to describe analytically are the behaviours of macroscopic quantities such as the one particle phase space density.

Studies of the dynamics of long-range interacting system (e.g. [34, 35] or for re-view [1]) – both numerical and theoretical – have lead to the conclusion that it is characterized broadly by two distinct regimes: starting from a generic initial condi-tion, a long-range system of many bodies evolves first, on a time scale (⌧mf or ⌧dyn)

characterizing the evolution under the mean field, towards a macroscopically station-ary state which does not coincide with the equilibria derived from the equilibrium thermodynamics. This early time process is often referred to as “violent relaxation”

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following the nomenclature originally given by Lynden-Bell for self-gravitating sys-tems [36]. These so-called quasi-stationary states are believed to describe notably globular clusters, galaxies [3], the red spots of Jupiter [4] and are reproduced in simulations (e.g.: for the HMF model [34], 1d self-gravitating system [18], 2d euler equations [37]) for various long-range systems. They are interpreted theoretically as stationary solutions of the Vlasov equation which we derive in the next section. This equation is a non linear equation for the single particle distribution function, and takes the mathematical form of a Boltzmann equation without collisional term. In the astrophysics community (see e.g. [3]) it is usually called the “collisionless” Boltzmann equation. It has been rigorously shown [38], for a certain class of poten-tial (with an exponent ↵  d 1 as we will see in chapter 2) that it gives a very good approximation, at a macroscopic level, of the N body dynamics at least on time scale ⌧V ⇠ ln(N)⌧mf

On a longer time scale diverging also with the particle number but (in general much longer than ⌧V), numerous long-range interacting systems have been observed

to relax to thermal equilibrium [1,28]. This second phase evolution is no longer de-scribed within the framework of the Vlasov equation and additional terms account-ing for “collisions” (or correlations, as we will discuss below) have to be included to describe it.

Derivation of the Vlasov equation

In this section we present a derivation of the Vlasov equation (see e.g. [39, 40]) through the BBGKY hierarchy and a short discussion of what this equation repre-sents.

– The BBGKY hierachy – We consider a system of N indistinguishable point particles of identical masses (or charges) m; we denote xi and vi respectively the

position and the velocity of the ith particle in a d dimensional space. At time t,

the information about an ensemble of realisations of the system is given by the N particle probability distribution function:

fN(xN, vN, t)= f. N(x1, ...xN, v1, ..., vN, t). (1.15)

It gives the probability of finding N particles in the infinitesimal volume element SN

i=1[xi, xi+ dxi][ [vi, vi+ dvi]in the 2dN dimensional space of configurations.

For s = 1, ..., N, we define the reduced s-particle phase space probability: fs(xs, vs, t)= f. s(x1, ...xs, v1, ..., vs, t) =

Z

PN(xN, vN, t)dx(N s)dv(N s), (1.16)

with the notation dz(N s) = dz.

s+1...dzN. The conservation of the probability give

the sum rule Z

fs(xs, vs, t)dxsdvs = 1, 8s = 1, ..., N, (1.17) where dxsdvs =. Qs

i=1dxidvi. As there is no sink or source in the space of

config-urations, a consequence of the divergence theorem (in 2dN-dimension) is that fN

obeys the continuity equation: @fN(xN, vN, t) @t + N X i=1  @xi @t · @fN(xN, vN, t) @xi +@vi @t · @fN(xN, vN, t) @vi = 0. (1.18)

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where · denotes the scalar product and we use the notation @

@z ⌘ rzfor the gradient.

Now if the N particles interact via a pair potential (|x y|), the Hamiltonian of this system is given by:

H = 1 2m N X i=1 vi2+ N X i=0 X j>i (|xi xj|). (1.19)

and we implicitly assume a regularization at r ! 0, if needed (we will come back to this crucial point). For a Hamiltonian system, the equations of motion are:

@xi @t = @H @mvi = vi (1.20) @vi @t = @H @xi = N X i6=j @ (|xi xj|) @xi .

Inserting then in Eq. (1.18), we obtain the Liouville equation: @fN(xN, vN, t) @t + N X i=1  vi· @fN(xN, vN, t) @xi (1.21) X j6=i @ (|xi xj|) @xi · @fN(xN, vN, t) @vi # = 0

Integrating Eq. (1.21) over velocities and position of N s particle we obtain the so called Bogoliubov-Born-Green-Kiriwood-Yvon (BBGKY) hierarchy and assuming that fN ! 0 when |v i|, |xi| ! +1, we have 8s = 1...N: @fs @t + s X i=1 vi· @fs @xi s X i6=j @ (|xi xj|) @xi · @fs @vi (1.22) = (N s) s X i=1 Z dxs+1dvs+1 @ (xi xs+1) @xi · @fs+1 @vi ,

where the factor (N s) comes from integrals over (assumed) indistinguishable particles. One sees that each equation for fs is coupled to the next one for fs+1.

This set of equations contain exactly the same information as the Liouville equation Eq. (1.21).

– The Vlasov equation – It takes the from of a closed equation for the equation for the single particle phase space density f1(x

1, v1). In order to obtain it, one needs

to close the BBGKY hierarchy at the level of f1 with assumptions that we discuss

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The first equation of the BBGKY hierachy (Eq. (1.22)) is: @f1(x 1, v1, t) @t + v1· @f1(x 1, v1, t) @x1 (1.23) (N 1) Z @ (|x1 x2|) @x1 · @f2(x 1, x2, v1, v2, t) @v1 dx2dv2 = 0.

Now let us decompose the two-body distribution function:

f2(x1, x2, v1, v2, t) = f1(x1, v1, t)f1(x2, v2, t) + g(x1, x2, v1, v2, t), (1.24)

where g is the connected or irreducible, two-point correlation function. To close the hierarchy i.e. to obtain an equation for the single particle distribution function f1

only, we simply assume that g(x1, x2, v1, v2, t) = 0. In other words, we assume that

the pair correlation and higher order correlations are negligible, or suppose what Boltzmann called molecular chaos in the context of a dilute gas. If we suppose the initial condition, indeed, satisfies, fN(x

1, x2, v1, v2, 0) = QNi=1f1(xi, vi, 0), one

may ask until which time t this assumption is a realistic description of the system. This property for a N body system to remains uncorrelated during its dynamical evolution is called propagation of chaos. With this assumption, and in the limit of large N, the last term can be rewritten:

(N 1) @ @v1 Z @ @x1 f2(x1, x2, v1, v2, t)dx2dv2 ' @f1 @v1 · @ ¯[f1](x 1, t) @x1 , (1.25) where the mean field potential ¯ is:

[f1](x1, t) = (N 1)

Z

(|x1 x2|)f1(x2, v2, t)dx2dv2. (1.26)

Let us now take the limit N ! +1 at fixed f1(x, v)(i.e. increasing the particle

density f = Nf1in phase space). We can obtain consistently an N independent limit

if we take (N 1) ⇠ Cst, in other words, rescale the coupling of the interaction by 1

N. This correspond to the Kac prescription, discussed earlier, in which the coupling

is scaled in order to obtain an extensive energy4. Indeed E ⇠ N2 ⇠ N when

⇠ 1

N. We obtain then the Vlasov equation in its dimensionless form:

@f1(x, v, t) @t + v· @f1(x, v, t) @x @ ¯[f1](x, t) @x · @f1(x, v, t) @v = 0, (1.27) with the mean field potential:

[f1](x, t) = Z

(|x x0|)f1(x0, v0, t)dx0dv0. (1.28)

For this equation, we recall that f1 represents the single particle probability

distri-bution normalised to unity.

4Here we assumed the system size is fixed, then this scaling is proportional to V as considered

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One may alternatively rewrite the equation in terms of the mass density in the phase space f(x, v, t) = Mf1(x, v, t). In the limit of large N with a potential scaling

as ⇠ 1

N and the mass m ⇠ 1

N, such that the total mass M = mN remains fixed,

we obtain the (dimensional) Vlasov equation: @f @t + v· @f @x @ [f ] @x · @f @v = 0, (1.29) with the mean field potential:

[f ](x) = 1 M

Z

(|x x0|)f(x0, v0, t)dx0dv0. (1.30) In this last expression Eq. (1.29), we see that the Vlasov equation is a continuous description of the particle dynamics in the limit of an infinite number of infinitely light particles. The mean field limit can thus be described as a fluid limit. It can be shown that the Vlasov dynamics has an infinite number of conserved quantities called Casimir invariants:

C[f ] = Z

H(f (x, v, t))dxdv, (1.31) where H is any function. The Vlasov equation admits an infinite number of sta-tionary solutions [1,41]. The continuous dynamics described by the Vlasov equation is a major subject of research in itself and also addressed in various different fields (astrophysics, plasma physics,..). Indeed, it is a complex non-linear equation and its resolution, just as the N body dynamics, is very costly numerically (and even more so because it is an equation for the 2d dimensional phase space).

This standard derivation, just given, of the Vlasov equation is not completely satisfactory: indeed, it remains quite unclear in what circumstances the essential approximation made of neglecting correlation will apply to a system in the class of Hamiltonian systems. It is known that for hard-core or short-range potential (e.g. van der Waals), the system is not well described by the Vlasov equation but instead by the Boltzmann equation with a collision term accounting for short-range correlations. Looking more carefully at the derivation, we have implicitly supposed, for example in Eq. (1.23), that integrals are not divergent. Also, for a system for which a short-scale regulation of the pair force is needed, the present derivation does not allows one to understand the effects of this regularization. We have simply stated that if we make certain approximations we obtain this equation, but have not justified the approximation with any rigour.

One of the aims of the chapter 2 of the thesis is precisely to improve our under-standing of the range of application of the Vlasov equation.

Violent relaxation: the establishment of QSS

As we have stated, studies of long-range systems show that they evolve generically through a phase of violent relaxation and organise in a QSS. Both this dynamical evolution and the QSSs are believed to be described by the Vlasov equation and the QSSs are stationary states of the latter5. As noted above, there are an infinite

5This statement “seems”,a priori, paradoxical because the macroscopic evolution described is

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number of stationary solutions of the Vlasov equation. The question then arise naturally of how the systems selects which stationary state it relaxes to and to what extent this does or does not depend on the detail of the initial conditions for the system. Indeed, it is known that the macroscopic distribution of the QSS reached depends on the initial conditions given to the system. As noted, the early time

Figure 1.4: Illustation of the “violent relaxation”. The plot shows a particles position in phases space at different time (N body simulations of 1d gravity [42])

relaxation can be studied within the framework of the Vlasov equation.

Lynden-Bell proposed a statistical approach to calculate the phase space distri-bution of QSSs [36] in the framework of the Vlasov equation that we briefly describe here. A detailed derivation can be found for example in [1,26,36,42]. The Lynden-Bell theory is similar to the usual Boltzmann statistics, but instead of working with the particles, Lynden-Bell studied the distribution of the phase space density level ⌘ at a microscopic level in phase space. This approach assumes, just as for Boltzmann statistics, the existence of ergodicity with respect to the phase space configuration accessible to the Vlasov dynamics i.e. consistentlty with its conserved quantities. The phase space is divided into P “macrocells” and each of them subdivided into ⌫ “microcells” of volume hd. The volume fraction occupied by the level ⌘ inside the

macrocell i is

⇢(x, v) = ni

⌫, (1.32)

where ni is the number of microcells inside a macrocell i occupied by the level ⌘. The

volume fraction is related to the distribution function by ⇢(x, v) = f(x, v)/⌘. Then with simple combinatorics, one can work out the number of possible “microstates” W (ni) and defined the entropy Slb = kBln(W (ni)). In the limit in which the

variaton of ⇢(x, v) between macrocells are infinitesimal, one can write the entropy as:

Slb = kB

Z

ddxddv

hd [⇢(x, v)⇢(x, v) + (1 ⇢(x, v))ln(1 ⇢(x, v))]. (1.33)

Then maximising the latter entropy with the constraint that the energy and the number of particle is conserved, one finds the following solution, for the case of a

equation induces a filamentation of the phase space. Looking at a macroscopic level, with time going on, the filaments, so thin, becomes indistinguishable. The time dependence is thus progressively transferred to small and smaller scales.

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single of a single level of density: flb(x, v) = ⌘ 1 + e (e(x,v) µ), (1.34) where e(x, v) = 1 2mv

2 + m ¯(x) is the single particle energy, and and µ are

the Lagrange multipliers related to the two constraints (interpreted as the “inverse temperature” and “the chemical potential”).

It has been found through numerical studies that the LB theory can provide a very accurate description of the relaxed QSS, in certain parts of the initial condition space of specific models: in the HMF [43], and in model with 2d vortices [44]. However more generally it is clear that the Lynden-Bell theory is inadequate, in particular for system starting from state very far from a QSS. This have been shown for the 1d self-gravitating system [18,42] and for the HMF [45]. In 3d gravity notably, an ubiquitous property of QSS is to have non-isotropic velocity distribution and even non isotropic spatial distribution arising from spherically symmetric initial condition and such state are not predicted by the theory of Lynden-Bell.

The main reason why the theory does not work for any initial condition is that it supposes a mixing and ergodicity in the phase space, and these two requirements are in general not verified by long-range systems. Other approaches to predict the property of QSS have been explored notably by [42]. The model constructed by the authors, have given, in various cases (HMF, 1d self-gravity and others), framework to understand well these states based on an analysis of the dynamics of the phase of violent relaxation. In summary, during violent relaxation, the mean field potential is characterised by quasi-periodic oscillations. These breathing modes have been notably shown within the framework of kinetic equation (beyond Vlasov), in the context of cold-atoms [5]. It is possible, therefore, for some particles to enter in resonance with the oscillations and gain large amounts of energy at the expense of the collective motion. This process is known as Landau damping. The Landau damping diminishes the amplitude of the oscillations and leads to the formation of a halo of highly energetic particles which surround the high density core [19]. The phenomenon of Landau damping, first described heuristically by Landau [46] in the context of plasma physics has been recently formulated rigorously in the famous work of Villani and Mouhot [47, 48] in particular limits.

Evolution of QSS: beyond Vlasov equation

The state attained by a long-range system is a quasi-stationary state because, for a large but finite N particle system, a second relaxation occurs, on a much longer time scale ⌧Reventually bringing the system toward the thermal equilibrium (if well

defined). The time scale, in general also diverges with N and its precise dependence varies with the model.

For 3d self-gravitating systems, Chandrasekahar [49] estimated the time scale for thermal relaxation to be ⌧R⇠ N/ln(N)⌧dyn. His calculation, based on an estimation

of the number of “collisions” a test particle has in the time it crosses the system (⇠ ⌧dyn). This estimate of the relaxation rate due to two-body collisions (for 3d

gravity) has been generalised to a broad class of pair interactions in dimensions d 2 in [50]. The latter calculation also shows that the main contribution to this

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rate for a long-range system is due to small angle deflections (or “soft” collisions) rather than due to “hard” collisions giving large deflections.

While the mean field dominates the trajectories of particles, many small deflec-tions lead eventually to significant deviadeflec-tions from the collective dynamics. The physics of such“collisions” is not included into the Vlasov equation describing only the collective effects, and additional term can be derived to account for them.

One common approach is to add Fokker-Plank to the Vlasov equation [3, 14]. To compute the diffusion coefficient, the microscopic detail of the mechanism of collision is needed with all the associated hypotheses underlying the calculation of Chandrasekar. This have been done notably in the context of 2d gravity in [51].

The Lennard-Balescu equation, coming from plasma physics, is another attempt to build such terms from a consistent derivation. To obtain it, one has to close the BBGKY hierarchy at the level of the two body correlation function f2. In order to do

so, one usually expands the pair density function as f2(z

1, z2) = f (z1)f (z2)+g(z1, z2)

as discussed above (we use the simpler notation zi = (xi, vi)and f = f1), and f3 as

f3(z1, z2, z3) = f (z1)f (z2)f (z3) + f (z1)g(z2, z3) (1.35)

+ f (z2)g(z1, z3) + f (z3)g(z1, z3) + h(z1, z2, z3).

with h the connected three-point correlation function.

Then assuming that h ⇠ 0 for large N, one obtains two coupled equation, closed in f and g. Then solving formally the equation for g, it is possible to obtain, for an homogeneous state (which is relevant in plasma physics but not for self-gravitating system), an expression for the additional term of the Vlasov equation in f which take into account the finite N effects. The additional term take finally the form of Fokker-Planck terms with a diffusion coefficient which is a functional of f. In practice to solve this equation is difficult and it remains unclear whether it correctly describe the evolution of relaxation of the system.

1.4 Long-range systems with stochastic perturbations

The Vlasov equation applies to strictly conservative systems in a micro-canonical framework, and the question inevitably arises of the robustness of QSSs, which are stationary solution of this equation, beyond this idealized limit.

In [52] the authors consider a long-range interacting system driven by exter-nal stochastic forces acting collectively on all the particles constituting the system. Given a long-range Hamiltonian H (Eq. (1.19)), they consider the following equation of motion: ˙qi = @H @pi and ˙pi = @H @qi ↵pi+p↵⇠(qi, t), (1.36)

where ⇠(q, t) is a statistically homogeneous Gaussian stochastic force field ( or “noise”) with a zero mean and without temporal correlation i.e

h⇠(q, t)i = 0 and h⇠(qi, t)⇠(qj, t)i = C(|qi qj|) (t t0), (1.37)

where ↵ controls the strength of the force and C is the (isotropic) correlation func-tion.

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Starting from a generalised Liouville equation for the N-particle distribution function fN (Eq. (1.15)), they derive a kinetic theory starting from a generalized

BBGKY hierarchy which takes the stochastic perturbation into account. Because the stochastic force is correlated in space it induces correlations on the system and, to treat this at a non-trivial level, the kinetic theory has to take into account the connected pair correlation g between particle in the derivation the kinetic equation. To obtain Lennard-Balescu equation, their generalised BBGKY hierarchy is closed at the level of g by neglecting the connected three-point and higher order correlation functions. They obtain thus a set of 2 equations, with both finite N effects and the stochastic effect. Considering the limit N 1 and N↵ 1, in which the characteristic time ⌧stoch = 1⌧dyn of the external perturbation is much shorter that

the one of the finite N effects ⌧f N ⇠ N⌧dyn, the author then treat the limit ↵ ⌧ 1.

The authors able to show, in particular, that detail balance is respected (and the system evolved to a Gaussian velocity distribution) if and only if the noise is white i.e. uncorrelated in space. In presence of correlated noise, the author shows that the system relaxes toward a non-equilibrium stationary state (NESS). The numerical simulations of the HMF model reported in these article give not only very good agreement with the kinetic theory developed for homogeneous state but also, for some particular value of the parameter of the perturbation, the system exhibits a very interesting bistable behaviour: the system switches abruptly and intermittently between a homogeneous and an inhomogeneous state.

In [53, 54] the authors consider a perturbation of the long-range dynamics with a diffusion process à la Langevin in order to attempt an operative description of a canonical ensemble for long-range interacting. There studies correspond to the one in [52] but for an uncorrelated noise. The author report also, in particular, that the system relaxes to non-equilibrium stationary state that are QSS.

In an other study [55] the HMF model is perturbed with a energy conserv-ing three-particle collision dynamics. At a given frequency, the velocities of three particles, chosen randomly within the whole system, are exchanged by a periodic permutation. The authors showed in this case the QSSs are “destroyed” by the stochastic dynamics, and that the perturbation makes the system relax faster the thermal equilibrium.

In the spirit of these studies, in the chapter 3 and chapter 4 we consider long-range systems subjected to various kinds of perturbations, different to those de-scribed above.

2 Self-gravitating systems

2.1 3d self-gravity

The gravitational N body problem is evidently the most notable example of a long-range interacting system. The problem can be relevant at different scales in astro-physics and in cosmology, e.g. planetary system, stars a globular cluster, stars in galaxies, galaxies in galaxy cluster. Considering such a system in an infinite space (or with periodic boundary conditions) is also a pertinent model in the context of the problem of non-linear structure formations in cosmology.

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Results from equilibrium statistical mechanics

The study of the thermodynamics of three dimensional self-gravitating systems is a specific example in which the small scale regulation of the divergence mentioned above is relevant. One also has to enclose the system of N particles of masses m in a spherical box of radius R. Then, in the mean field limit N ! +1 with ER/GM2 = Cst where M is the mass of the whole system and E the energy or

N ! +1 with GMm/R = Cst with = (k

BT ) 1 the inverse temperature), it

is possible to determine the phase diagram in the canonical and micro-canonical ensemble. A result it is important to underline is that the phase diagram obtained depends on the small scales cut-off introduced to regularise the potential at r = 0 [56]. More precisely it depends not only on the nature of this cut-off (one can use a softened potential [57] or consider fermionic particles [24]) but also on its value. Without regularisation, nevertheless the problem can be solved formally using the mean field limit [14] as there are, for example, local maxima of entropy which are well defined independently of the cut-off. A famous result of the calculation is that we obtain a caloric curve spiralling around a point. It presents then a non concave region and the ensembles are not equivalent in general. With a short-scale regularisation, this is also true and lead to a very rich thermodynamics [56] [24].

When the energy in the micro-canonical ensemble or the temperature in the canonical ensemble drops below a certain critical value Ec or Tc, respectively, the

corresponding thermodynamic potentials undergo a discontinuous jump [14, 56]. If no short-range cut-off is introduced, the discontinuous jump is infinite and the en-tropy and free energy diverge, and then no extrema can be found. This makes all normal (non-singular) states of the self-attractive system metastable with respect to such a collapse; the collapse energy Ec is in fact an energy below which the

metastable states cease to exist [57]. Above this energy, isothermal sphere type solutions [14] exist. If, on the other hand, a short-range cutoff is introduced, the entropy and free energy jumps are finite. In this case, as a result of the collapse, the system goes into a non-singular state with a dense core and a sparse halo. The precise nature of the state depends on the details of the short-range behaviour of the potential [24]. There is an energy Ept for which both core-halo and

isother-mal systems have the same entropy; above this energy the collapsed state becomes metastable and at some higher energy it ceases to exist. It is possible to regard the energy Ept as that where a true (first order) phase transition occurs. The value of

Ept, and in general the form of the entropy-energy curve, is highly sensitive to the

details of the short-range cutoff [58].

These results thus shows a rich variety of equilibrium states. However both as-trophysical observation and numerical simulations show that, as we have discussed above, a self-gravitating system is rapidly trapped into a QSS (in astrophysics lit-erature they more usually referred to as “collisionless equilibria”). While numerical simulations show that the these states evolve on time scales in very good agreement with the predictions of Chandrasekhar described above [59–61], whether the equi-libria derived from the mean field limit of equilibrium thermodynamics are actually ever attained by such system remains, to our knowledge, an unanswered question.

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The Virial theorem and virialization.

As explained above a system starting from a phase space distribution which is not a time independent solution of the Vlasov equation undergoes a rapid evolution toward a Vlasov stable state (QSS).

One of the most noteworthy properties of such state is that they are so-called virial equilibria. This results is based on the “virial theorem” which we briefly discuss now for the self-gravitating case (but which can be generalised for any potential). We consider a system of self-gravitating particles in 3d interacting only through an exact 1/r potential. We introduce the inertia tensor [3]:

Iµ⌫ = N

X

i=1

mxi,µxi,⌫, (1.38)

where xi,µ is the µth component of the ith particle. The second time derivative of

this quantity is:

¨ I = m

N

X

i=1

(¨xi,µxi,⌫ + xi,µx¨i,⌫+ 2 ˙xi,µ˙xi,⌫) . (1.39)

Inserting the equations of motion of the particles in the last expression, one easily obtains ¨ I = 2m N X i=1 vi,µvi,µ Gm2 N X i=1

(xi,µ xi,µ)(xj,⌫ xi,⌫)

|xi xj|3

. (1.40) Taking the trace of the last expression one recognises that the first term on the right hand side is four times the kinetic energy and the second twice the potential energy. One thus obtains the Lagrange identity:

1

2I = 2K + U¨ (1.41) If now we assume that the system is, at a macroscopic level, in a stationary state, one obtains:

2K + U = 0 (1.42)

Thus, up to fluctuations due to finite particle number, we expect any such system to obey the relation (1.42). It can alternatively be directly derived from the Vlasov equation assuming only stationarity of f. We will illustrate this below for the case of a 1d self-gravitating system. It is straightforward also to generalise this result to any potential 1/r↵ and to a system confined in a box, associated with a pressure:

2K + ↵U = 3P V 8↵ 6= 0 (1.43) where P is the pressure and V the volume of the system.

For a system in thermal equilibrium at temperature T , K = 3

2N kBT, or more

generally one can defined a kinetic temperature in this way (by 3

2N kBT = hv 2i)

Then, if the pressure term can be neglected (e.g. in any QSS of an unbound self-gravitating system), we have, for the gravitational case: 2K + U = 0, thus E = K + U = K = 32N kBT, implying dEdT = kB32N. But dEdT is just, for a system in

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above. Thus when a self-gravitating system loses energy it heats up. By losing heat, the system grows hotter and continues to radiate energy. This is the mechanism that prevails in the internal region of stars [24]. For instance, at a stage where no more nuclear fuel is available, the core contracts and becomes hotter, giving its energy to the outer part which expands and becomes colder.

2.2 1d self-gravity

We introduce here briefly the 1d self-gravitating system, which is a paradigmatic model for long-range interacting systems. Besides the fact that it provides a natural case to study as a toy model for 3d gravity, this 1d model has the very nice feature that the particles trajectories can be analytically integrated between crossing which means that they can be simulated “exactly” as describe in detail in chapter 3. Definition of the model

We consider a system of N particles of identical masses m = 1 moving in 1d and interacting through the gravitational pair interaction which obey the 1d Poisson equation:

@x2 (x) = 2g (x), (1.44) with g the coupling constant. The model can be also seen as infinite self-gravitating sheets in three dimensions of surface mass density ⌃, which lead to the identification has g = 2⇡G⌃ with G the (3d) gravitational constant.

Then, in a system of N such particles, the force exerted on a particle i by all the other is: Fi = g N X j6=i sgn(xj xi) , (1.45)

which can be written as

Fi = g [N>(xi) N<(xi)] (1.46)

where N> ( N<) the number of particle on the left (right) of the particle i. Thus

the force is constant other than when particles cross.

We note further that compared to 3d gravity, 1d gravity differs notably in that (1) the force (and potential) is regular at r = 0 and thus no short-scale regularization is required and (2) the potential diverges at large separation and is therefore a confining potential. No box is therefore necessary to treat the equilibrium thermodynamics. Virial theorem

In this section we derive the virial theorem for 1d gravity starting from the Vlasov equation. Integrating the 1d Vlasov equation over v, and using the assumption f (x, v, t) ! 0 for v, x ! ±1, one obtains the continuity equation:

@⇢(x, t) @t +

@ (⇢(x, t)v(x, t))

@x = 0. (1.47)

Now, if we multiply the Vlasov-Poisson equation by x and v and integrate with respect the two variables over the whole space we obtain

Z xv@f @tdxdv + Z xv2@f @xdxdv Z xv@ @x @f @vdxdv = 0 . (1.48)

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Considering first the integration over v, and integrating by parts the last term, this last equation is rewritten:

Z x@ (⇢(x, t)v(x, t)) @t dx + Z x@(⇢(x, t)v 2(x, t)) @x dx + Z x⇢(x, t)@ @xdx = 0. (1.49) Integrating by parts and using the continuity equation (1.47), the first term gives:

1 2 d dt Z x2@⇢(x, t)v(x, t) @x = 1 2 d2 dt2 Z x2⇢(x, t) = 1 2 d2I(t) dt2 . (1.50)

with I the moment of inertia. The second term is Z

x@(⇢(x, t)v2(x, t)) @x dx =

Z

⇢(x, t)v2(x, t) = 2K(t). (1.51)

Finally, the third term equals may be written as Z x⇢(x, t)@ ¯(x) @x dx = 1 M Z Z ⇢(x, t)gmx(x x 0) |x x0| ⇢(x0, t)dxdx0. (1.52)

Noting that the symmetric part, by exchange x $ x0, of = x(x x0)

|x x0| is Sym = 1

2|x x0| and as ⇢(x, t)⇢(x0, t)is symmetric under this exchange, only the symmetric

part survives in the integral and the third term equals the mean field potential energy:

1 2M

Z Z

⇢(x0, t)gm|x x0|⇢(x, t)dx0dx = U (t). (1.53) Collecting all terms one obtains the Lagrange indentity

1 2

d2I

dt2 = 2K(t) U (t). (1.54)

Hence for stationary states of the Vlasov equation, i.e. for QSS, we obtain the virial theorem for 1d gravity: 2K = U. In this case a virialised state verify E = 3K =

3

2U = C

ste. Further we note the specific heat is always positive, and indeed, it turns

out that as we saw, the thermodynamic ensembles are equivalent. Thermodynamics

As we have noted, this model does not present the small scale divergence problem; moreover, as we noted, it is not necessary to enclose the system in a box. The equilibrium thermodynamics has been completely solve by Rybicki in 1971 [20] and reproduced in greater detail in [26]. Remarkably, the calculation give an exact expression, even for a finite N, of the equilibrium solution in both the canonical and micro-canonical ensemble. The Hamiltonian of the system is

H(x, v) = K(v) + U (x) = N X i 1 2mv 2 i + gm2 2 N X j=1 N X k6=j |xj xk|; (1.55)

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where x = x1, ...xN and v = v1, ..., vN. In this calculation, we start from the

definition of the equilibrium distribution in the canonical ensemble fc(x, v) = (zN !) 1 Z Z dxNdvN (¯x) (¯v)e H(v,x) 1 N N X i=1 (x xi) (v vi) , (1.56)

where (¯x) (¯v) is the constraint on the center of mass to remains fixed at the ori-gin, and z the corresponding partition function. The factor N! arises because we assumed indistinguishable particles. With this assumption the distribution function

1 N

PN

i=1 (x xi) (v vi)can be replaced by (p pN) (x xN)under the integral.

Hence, an important consequence, is that the equilibrium solution is separable: fc(x, v) = ⇢c(x)⇥c(v), (1.57) with ⇢c(x) = (QN !) 1 Z ddx (¯x)e U (x) (x xN), (1.58) and ⇥c(v) = (R) 1 Z ddv (¯v)e K(v) (v vN), (1.59)

where Q and R are the corresponding partition functions which normalise the two distributions with the relation z = QR. And then by direct calculation, using Fourier transform and complex analysis one can show that

⇥c(v) = s N 2⇡m(N 1)e B mv2 2(N 1), (1.60) and ⇢c(x) = N gm2 N X j=1 AN j e N gm 2j|x| , (1.61) with ANj = j( 1)j+1[(N 1)!]2 (N 1 j)!(N 1 + j)!. (1.62) The calculation of the equilibrium solution in the micro-canonical ensemble is then obtained performing an inverse Laplace transform of the canonical solu-tions [20]. Then taking the limit N ! +1 at fixed mass M and energy E using the characteristic momentum and length scale: 2 = 4m2E

3M and ⇤ = 4E 3gM2, one obtains: feq(x, v) = 1 2p⇡ 1 ⇤sech 2⇣ x ⇤ ⌘ e( v)2 (1.63) Dynamics

The dynamics of 1d self-gravitating system is qualitatively typical of that of any long-range interacting system as we have outlined it above. In particular, as we will detail and illustrate with results from our simulation in chapter 3, the system reaches first a given QSS on a time scale of order ⌧dyn after a period of violent relaxation.

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much longer times scales, proportional to N in units of ⌧dyn, these QSS relax to the

thermal equilibrium (given by Eq. (1.63) above). One notable feature is that this relaxation is very slow, in the sense that ⌧relax ' zN⌧dynwhere z is a numerical factor

of order 102 103 which depends on the QSS [28]. The calculation of Chandrasekhar

for 3d is not generalizable to this case as there are no real collisions in 1d because particles just cross on another and indeed the mechanism for this relaxation to thermal equilibrium remains an open problem for this system.

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Finite N corrections to Vlasov

dynamics

We saw in the previous chapter that one of the most interesting feature of the dynamics of a long-range interacting system it that they are trapped into QSS. A basic question is whether the appearance of these out of equilibrium stationary states — and more generally the validity of the Vlasov equation to describe the system’s dynamics — applies to the same class of long-range interactions as defined by equilibrium statistical mechanics, or only to a sub-class of them, or indeed to a larger class of interactions. In short, in what class of systems can we expect to see these quasi-stationary states? Are they typical of long-range interactions as defined canonically? Or are they characteristic of a different class?

To answer these questions requires establishing the conditions of validity of the Vlasov equation, and specifically how such conditions depend on the two-body in-teraction. In the literature there are, on the one hand, some rigourous mathematical results establishing sufficient conditions for the existence of the Vlasov limit. It has been proven notably [38, 62, 63] that the Vlasov equation is valid on times scales of order ⇠ logN times the dynamical time, for strictly bound pair potentials decaying at large separations r slower than r (d 2). On the other hand, both results of

numer-ical study and various theoretnumer-ical approaches, based on different approximations or assumptions, suggest that much weaker conditions are sufficient, and the timescales for the validity of the Vlasov equation can be much longer. In the much studied case of gravity, notably, a treatment originally introduced by Chandrasekhar, [49] and subsequently refined by other authors (see e.g. [35,59,64,65] in which non-Vlasov ef-fects are assumed to be dominated by incoherent two body interactions gives a time scale ⇠ N/logN times the dynamical time for the validity of the Vlasov equation, at least close to stationary solutions representing quasi-stationary states, and this in absence of a regularisation of the singularity in the two-body potential. Theoretical approaches in the physics literature derive the Vlasov equation and kinetic equa-tions describing correcequa-tions to it (for a review see [1,66]) either within the framework of the BBGKY hierarchy [40], as we saw in the previous chapter, or starting from the exact Klimontovich equation for the N body system [67, 68]. These approaches are both widely argued (e.g. see e.g. [1, 2, 66, 69, 70], to lead generally to lifetimes of quasi-stationary states of order ⇠ N times the dynamical time for any softened pair potential , except in the special case of spatially homogeneous quasi-stationary states one dimension.

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In this chapter, we address the question of the validity of Vlasov dynamics using an approach starting from the exact Klimontovich equation. Instead of considering, as is often done (see e.g. [1,2,69] , an average over an ensemble of initial conditions to define a smooth one particle phase density, we follow an approach (described e.g. in [71]) in which such a smoothed density is obtained by performing a coarse-graining in phase space. This approach gives the Vlasov equation for the coarse-grained phase space density when certain terms are discarded. We study how the latter “non-Vlasov” terms depend on the particle number N, on the scales introduced by the coarse-graining. In particular we develop this study analyzing the dependence on the large and small distance behavior of the two body potential. Our analysis leading to the scaling behaviours of these terms is based only one very simple — but physically reasonable — hypothesis that we can neglect all correlations in the (microscopic) N body configurations other than those coming from the mean (coarse-grained) phase space density.

The main physical result we highlight is that, under this simple hypothesis, the coarse-grained dynamics of an interacting N-particle system shows a very different dependence on the pair interaction at small scale depending on how fast the interac-tion decays at large distances: for interacinterac-tions of which the pair force is integrable at large scales the coarse-grained N body dynamics is highly sensitive to how the poten-tial is softened at much smaller scales, while for pair forces which are non-integrable the opposite is true. Correspondingly, while the Vlasov limit may be obtained for any pair interaction which is softened suitably at small scales, the conditions on the short-scale behaviour of the interaction are very different depending on whether its large scale behaviour is in one of of these two classes. This result provides a more rigorous basis for a “dynamical classification" of interactions as long-range or short-range, which has been introduced on the basis of simple considerations of the probability distribution of the force on a random particle in a uniform particle dis-tribution in [72], and found also in [50] to coincide with a classification based on the dependence on softening of collisional time scales using a generalisation of the analysis of Chandrasekhar for the case of gravity.

The chapter is organized as follows. In the next section we derive the equation for the coarse-grained phase space density and write in a simple form the non-Vlasov terms our subsequent analysis focusses on. In section 2 we first explain our central hypothesis concerning the N-body dynamics, and then apply it to evalu-ate the statistical properties of the non-Vlasov terms. In the following section we then determine the scaling behaviours of these expressions, i.e., how they depend parametrically on the relevant parameters introduced, and in particular on the two parameters characterising the two body interaction — its large scale decay and the scale at which it is softened. In section 4 we use these expressions to identify the dominant contributions to the non-Vlasov terms, which turn out to differ depending on how rapidly the interaction decays at large scales. In the following section we present more complete exact results for the one dimensional case and the comparison with a simple numerical simulation. We then summarize our results and conclusions, discussing in particular the central assumptions and the dependence of our findings on them.

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1 A Vlasov-like equation for the coarse-grained phase

space density

We summarize a standard approach used to justify the validity of the Vlasov equa-tion for long-range interacting systems alternative to the one described in the chap-ter 1 (BBGKY). The approach involves using a coarse-graining, in phase space, of the full N body dynamics and leads to an evolution equation for the coarse grained phase space density which consists of the Vlasov terms, plus additional terms. This equation, and the specific form of the non-Vlasov terms we derive, is the starting point for our analysis in the subsequent sections. We follow closely at the beginning the presentation and notation of [71].

We consider a d-dimensional system of N particles of identical mass m = 1 interacting only through the a generic two body force, denoting g(x) the force on a particle at x exerted by another one at the origin.

At any time t, the N particles have phase space positions which we denote {(xi, vi)}i=1..N, and the microscopic (or fine-grained, or Klimontovich) one particle

phase-space density is simply the distribution fk(x, v, t) =

N

X

i=1

(x xi(t)) (v vi(t)). (2.1)

Likewise the microscopic one particle density distribution in coordinate space is nk(x) = Z fkddv = N X i=1 (x xi), (2.2)

The full evolution of the N body system can be written in the form of the so-called Klimontovich equation for the microscopic phase space density:

@fk @t + v @fk @x + F[nk](x) @fk @v = 0, (2.3) where F[nk](x) = Z ⌦ g(x x0)nk(x0)ddx0 = N X i=1 g(x xi), (2.4)

is the exact force at point x (due to all particles). The detail derivation of this equation is given in [1], this equation is exact and contain the same information than the Hamilton’s equation or the Liouville equation (cf. chapter 1). The only assumption made in deriving this equation from the equations of motion of the individual particles is that the force g(x) is bounded as x ! 0.

Introducing a top-hat window function W (z = z1, . . . , zd),

W (z) = ⇢

1, if|z| < 1,

0, otherwise, (2.5) we define the coarse-grained phase space density:

f0(x, v, t) = Z ddx0 d x ddv0 d v W ✓ x x0 x ◆ W ✓ v v0 v ◆ fk(x0, v0, t), (2.6)

Figure

Figure 1.1: Schematic picture of a system partitioned into two parts, in red is the boundary between the two subsystems.
Figure 1.2: Example of a concave entropy-energy curve (left panel) and its Legendre- Legendre-Fenchel transform (right panel): the free-energy temperature curve.
Figure 1.3: Sketch of a non concave entropy S verus the energy E (left panel), its Legendre transform S ⇤ ( ) = F ( ) (middle panel), and the Legendre-Fenchel
Figure 1.4: Illustation of the “violent relaxation”. The plot shows a particles position in phases space at different time (N body simulations of 1d gravity [42])
+7

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